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On chaotic dynamics of nonanalytic quadratic planar maps
Kutnjak, Milan
In this article we consider the dynamics of a special example of nonanalytical homogeneous quadratic map in the plane. It is well-known (5) that to every homogenous quadratic map in ▫$\mathbb R^2$▫ ...one can associate a commutative (possibly nonassociative) algebra. The correspondence between quadratic maps and algebras is one-to-one. Classical Julia sets actually originate from studying the iteration of the complex squaring map. However, the algebra of complex numbers is just one of many nonisornorphic commutative algebras existing in ▫$\mathbb R^2$▫. Concerning the algebraic properties like idempotents and nilpotents we searched for the algebra which is most similar to the algebra of complex numbers. The associated discrete dynamical system in this particular algebra exhibits similar behavior on the boundary of basin of attraction of the origin as the cornplex-squaring map on the classical Julia set. We used a one-to-one projection to the unit circle to prove the chaotic properties of the invariant set.
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